If , then is recursive iff both A and B are recursive.
Let and let n ;SPMgt; 1. Prove that if B is recursive
then the predicate given by given by
is decidable.
Let . Prove that A is recursive iff
is recursive.
For any , let . Show that
is r.e for all a. Does the enumeration
include all r.e. sets?
Show that the set is r.e.
Show that there are total computable functions k,l such that for
every x, and .
Suppose that A is an r.e set. Show that the sets
and are both r.e. Show that
is not necessarily r.e. as follows. For ant t let
. Show that for ant
t, is recursive; moreover and
.
Let f be a unary computable function, and suppose that
, and let . Prove that g is
computable iff A is r.e.
Let f is a unary function. Prove that f is computable iff the
set is r.e.
Let A be an inifinite r.e. set. Show that A can be enumerated without
repetitions by a total computable function.
Which of the following sets are recursive? Which are r.e.? Which
have r.e. complement?
(m is fixed)
Let and let n ;SPMgt; 1. Prove that id B is r.e then
the predicate given by
is partially decidable.
Let . Prove that A is r.e. iff
is r.e..
Prove that is r.e. iff or there
is a total computable function
such that A = Ran(f). (By a total computable function
from to we mean an n-tuple
where each is a unary computable
function and .)
Suppose that f is a total computable function, A is a recursive
set and B is a recursively enumerable set. Show that
is recursive and that f(A), f(B) and are r.e. but
not necessarily recursive. What extra information can be obtained about these sets
if f is a bijection?
Use Rice-Shapiro's theorem to show that the following problems are not
partially decidable:
` '
` '
` '
` '
` '
Prove Rice's theorem from the Rice-Shapiro's theorem.
[Hint: Suppose that ` ' is decidable. Then both
and satisfy the conditions of Rice-Shapiro:
consider the cases and .].